Normalized Iterative Hard Thresholding for Matrix Completion

Matrices of low rank can be uniquely determined from fewer linear measurements, or entries, than the total number of entries in the matrix. Moreover, there is a growing literature of computationally efficient algorithms which can recover a low rank matrix from such limited information; this process is typically referred to as matrix completion. We introduce a particularly simple yet highly efficient alternating projection algorithm which uses an adaptive stepsize calculated to be exact for a restricted subspace. This method is proven to have near-optimal order recovery guarantees from dense measurement masks and is observed to have average case performance superior in some respects to other matrix completion algorithms for both dense measurement masks and entry measurements. In particular, this proposed algorithm is able to recover matrices from extremely close to the minimum number of measurements necessary.